{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 简单理解 Density Fitting RMP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 创建时间：2019-10-16"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这一节的内容承接上一节对 DF RHF 的理解，只是简单地近一步得到 RMP2 的能量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "\n",
    "import numpy as np\n",
    "import scipy\n",
    "from functools import partial\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from pyscf import gto, scf, mp, df\n",
    "\n",
    "np.einsum = partial(np.einsum, optimize=[\"greedy\", 1024 ** 3 * 2 / 8])\n",
    "np.set_printoptions(8, linewidth=150, suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们使用的体系仍然是不对称的双氧水分子："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "mol = gto.Mole()\n",
    "mol.atom = \"\"\"\n",
    "O  0.0  0.0  0.0\n",
    "O  0.0  0.0  1.5\n",
    "H  1.0  0.0  0.0\n",
    "H  0.0  0.7  1.0\n",
    "\"\"\"\n",
    "mol.basis = \"def2-tzvp\"\n",
    "mol.verbose = 0\n",
    "mol.build()\n",
    "\n",
    "natm = mol.natm\n",
    "nao = nmo = mol.nao\n",
    "nocc = mol.nelec[0]\n",
    "nvir = nmo - nocc\n",
    "so, sv, sa = slice(0, nocc), slice(nocc, nmo), slice(0, nmo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PySCF RMP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PySCF 普通 RMP2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在我们跑 DF RMP2 前，我们先跑一下普通的 RMP2，以作对比。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5378068680460026"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scf_normal = scf.RHF(mol).run()\n",
    "mp2_normal = mp.MP2(scf_normal).run()\n",
    "mp2_normal.e_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PySCF DF RHF"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在 PySCF 中，即使自洽场已经经过 DF 优化过，也不需要特别地对 MP2 的代码作改变："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5370249637959694"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scf_df = scf.RHF(mol).density_fit().run()\n",
    "mp2_df = mp.MP2(scf_df).run()\n",
    "mp2_df.e_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在相关能上，两者的结果还是稍有差距的。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0007819042500332163"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp2_df.e_corr - mp2_normal.e_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于 PySCF 而言，一种更好的做法是，在自洽场过程中使用对 J/K 积分优化较好的 `jkfit` 型 DF 基组，而在 MP2 或后自洽场中使用对 MP2 优化较好的 `ri` 型 DF 基组："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5377072684090778"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scf_df = scf.RHF(mol).density_fit(auxbasis=\"def2-tzvp-jkfit\").run()\n",
    "mp2_df = mp.MP2(scf_df).density_fit(auxbasis=\"def2-tzvp-ri\").run()\n",
    "mp2_df.e_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在相关能的误差就相对来说小了不少。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.959963692485196e-05"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp2_df.e_corr - mp2_normal.e_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 重现 PySCF DF RMP2 相关能"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DF 基组定义与必要张量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们首先生成计算 DF RMP2 所需要的张量：\n",
    "\n",
    "* `V_df_mp2` $V_{\\mu \\nu, P}^\\mathrm{MP2}$：经过 Cholesky 处理的三中心双电子张量\n",
    "\n",
    "* `C`, `Co`, `Cv` $C_{\\mu p}$：分子轨道系数 `C`，以及其在占据轨道 (`Co`)、非占轨道 (`Cv`) 的分割\n",
    "\n",
    "* `D_iajb` $D_{ij}^{ab}$：即 $\\varepsilon_i - \\varepsilon_a + \\varepsilon_j - \\varepsilon_b$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们指出，这里计算的 `V_df_mp2` $V_{\\mu \\nu, P}^\\mathrm{MP2}$ 是由 RI 基组 `def2-tzvp-ri` 产生；与上一篇文档提到的由 `def2-tzvp-jkfit` 所生成的 $V_{\\mu \\nu, P}$ 是不同的。因此，我们需要使用对应的 `def2-tzvp-ri` 基组来产生 DF 分子类 `auxmol`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "auxmol = mol.copy()\n",
    "auxmol.basis = \"def2-tzvp-ri\"\n",
    "auxmol.build()\n",
    "\n",
    "nao_df = auxmol.nao"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "int2c2e = auxmol.intor(\"int2c2e\")\n",
    "int2c2e.shape\n",
    "int3c2e = df.incore.aux_e2(mol, auxmol)\n",
    "\n",
    "int2c2e_half = scipy.linalg.cholesky(int2c2e, lower=True)\n",
    "V_df_mp2 = scipy.linalg.solve_triangular(int2c2e_half, int3c2e.reshape(-1, nao_df).T, lower=True)\\\n",
    "               .reshape(nao_df, nao, nao).transpose((1, 2, 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "C, e = scf_df.mo_coeff, scf_df.mo_energy\n",
    "Co, Cv = C[:, so], C[:, sv]\n",
    "eo, ev = e[so], e[sv]\n",
    "D_iajb = eo[:, None, None, None] - ev[None, :, None, None] + eo[None, None, :, None] - ev[None, None, None, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ERI 积分转换"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "MP2 中最为耗时的一步是原子轨道 4c2e 的 ERI 积分 $(\\mu \\nu | \\kappa \\lambda)$ 转换为分子轨道 $(ia|jb)$ 的过程。我们在 DF MP2 中仍然没有避开生成 $(ia|jb)$，但我们避开了直接生成 $(\\mu \\nu | \\kappa \\lambda)$；恰恰是这一步成为 DF MP2 的优势。其具体的过程大致如下。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "首先，我们生成分子轨道下的三中心积分 $V_{ia, P}^\\mathrm{MP2}$ `V_df_ia` ($O(N^4)$ 复杂度)：\n",
    "\n",
    "$$\n",
    "V_{ia, P}^\\mathrm{MP2} = V_{\\mu \\nu, P}^\\mathrm{MP2} C_{\\mu i} C_{\\nu p}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "V_df_ia = np.einsum(\"uvP, ui, va -> iaP\", V_df_mp2, Co, Cv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "尽管我们使用了与 `jkfit` 不同的基组，但我们仍然认为 $(\\mu \\nu | \\kappa \\lambda) \\simeq V_{\\mu \\nu, P}^\\mathrm{MP2} V_{\\kappa \\lambda, P}^\\mathrm{MP2}$ 近似成立。因此，我们可以很容易地给出 ($O(N^5)$ 复杂度)\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "(ia|jb) &= C_{\\mu i} C_{\\nu a} (\\mu \\nu | \\kappa \\lambda) C_{\\kappa j} C_{\\lambda b} \\\\\n",
    "&\\simeq C_{\\mu i} C_{\\nu a} V_{\\mu \\nu, P}^\\mathrm{MP2} V_{\\kappa \\lambda, P}^\\mathrm{MP2} C_{\\kappa j} C_{\\lambda b} \\\\\n",
    "&\\simeq V_{ia, P}^\\mathrm{MP2} V_{jb, P}^\\mathrm{MP2}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "eri_df_iajb = np.einsum(\"iaP, jbP -> iajb\", V_df_ia, V_df_ia)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当然，我们也可以使用下述代码，一次性地从 $V_{\\mu \\nu, P}^\\mathrm{MP2}$ 生成 DF 近似的 $(ia|jb)$ `eri_df_iajb`，而结果和效率没有任何影响："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(\n",
    "    np.einsum(\"ui, va, uvP, klP, kj, lb -> iajb\", Co, Cv, V_df_mp2, V_df_mp2, Co, Cv),\n",
    "    eri_df_iajb\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最后，我们通过图像指出，DF 近似的转换后的分子轨道 ERI 与普通的分子轨道 ERI 非常接近。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "eri = mol.intor(\"int2e\")\n",
    "eri_iajb = np.einsum(\"ui, va, uvkl, kj, lb -> iajb\", Co, Cv, eri, Co, Cv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "bins =np.arange(-12, 1., 0.1)\n",
    "histo_eri = np.histogram(np.log10(np.abs(eri_iajb).ravel() + 1e-20), bins=bins)\n",
    "histo_df_eri = np.histogram(np.log10(np.abs(eri_df_iajb).ravel() + 1e-20), bins=bins)\n",
    "histo_dev = np.histogram(np.log10(np.abs(eri_df_iajb - eri_iajb).ravel() + 1e-20), bins=bins)\n",
    "\n",
    "prob_eri = histo_eri[0] / eri_iajb.size\n",
    "prob_df_eri = histo_df_eri[0] / eri_iajb.size\n",
    "prob_dev = histo_dev[0] / eri_iajb.size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
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       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width, fig.canvas.height);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"\" width=\"640\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa3bc1348d0>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.plot(histo_eri[1][:-1], prob_eri, label=\"Original MO ERI\", linestyle=\"-.\")\n",
    "ax.plot(histo_df_eri[1][:-1], prob_df_eri, label=\"DF MO ERI\", linestyle=\":\")\n",
    "ax.plot(histo_dev[1][:-1], prob_dev, label=\"Deviation\")\n",
    "\n",
    "ax.set_xlabel(\"$\\mathrm{log}_{10} \\mathrm{MO\\, ERI}$\")\n",
    "ax.set_ylabel(\"Probability Distribution\")\n",
    "ax.set_title(\"MO ERI Distribution Status\")\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上图中，蓝色线是普通的 MO ERI，它与橙色线几乎重合，意味着两者数值上极为相近。同时，作为两者之差的绿色线的分布明显在更小的数量级上。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 积分转换的复杂度简单分析"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上述的复杂度是不难分析的。尽管 DF 与普通 MP2 所进行的积分转换都是 $O(N^5)$ 的复杂度；但细致地分析会发现，普通 MP2 的复杂度是 $O(n_\\mathrm{occ} n_\\mathrm{AO}^4)$，而 DF 的复杂度是 $O(n_\\mathrm{occ}^2 n_\\mathrm{vir}^2 n_\\mathrm{aux})$，两者的计算次数是远不相同的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们有 `einsum_path` 的工具；通过该工具的 `Optimized FLOP count` 输出，我们能知道实际上 NumPy 将会计算多少次浮点计算。同时，`Largest intermediate` 会告诉我们在当前的张量缩并路径下，内存中需要使用的最大的中间张量大小。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下述的缩并过程是普通 MP2 的分子轨道基组 ERI 计算过程\n",
    "\n",
    "$$\n",
    "(ia|jb) = C_{\\mu i} C_{\\nu a} (\\mu \\nu | \\kappa \\lambda) C_{\\kappa j} C_{\\lambda b}\n",
    "$$\n",
    "\n",
    "若使用朴素的 (Naive) 张量乘积 ($O(N^8)$ 复杂度)，那么需要进行 $5.131 \\times 10^{13}$ 次浮点计算。我们知道通常的 MP2 是 $O(N^5)$ 复杂度；其对应的浮点计算次数在当前的双氧水体系下是 $7.137 \\times 10^8$ 次。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  ui,va,uvkl,kj,lb->iajb\n",
      "         Naive scaling:  8\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  5.131e+13\n",
      "  Optimized FLOP count:  7.137e+08\n",
      "   Theoretical speedup:  71892.409\n",
      "  Largest intermediate:  3.647e+06 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   5               uvkl,ui->iklv                      va,kj,lb,iklv->iajb\n",
      "   5               iklv,kj->ijlv                         va,lb,ijlv->iajb\n",
      "   5               ijlv,va->ijal                            lb,ijal->iajb\n",
      "   5               ijal,lb->iajb                               iajb->iajb\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"ui, va, uvkl, kj, lb -> iajb\", Co, Cv, mol.intor(\"int2e\"), Co, Cv, optimize=True)[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "而在 DF 近似于优化下，\n",
    "\n",
    "$$\n",
    "(ia|jb) \\simeq C_{\\mu i} C_{\\nu a} V_{\\mu \\nu, P}^\\mathrm{MP2} V_{\\kappa \\lambda, P}^\\mathrm{MP2} C_{\\kappa j} C_{\\lambda b}\n",
    "$$\n",
    "\n",
    "其对应的浮点计算次数则是 $1.920 \\times 10^8$。可以看到，DF 速度提升大约 3.72 倍。除此之外，积分转换对内存空间的需求还是普通 MP2 的 1/10.66 倍，因此可以说 DF MP2 是在损失比较小的精度的前提下，同时提升计算效率与内存利用，是一个很好的 Balance。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Complete contraction:  ui,va,uvP,klP,kj,lb->iajb\n",
      "         Naive scaling:  9\n",
      "     Optimized scaling:  5\n",
      "      Naive FLOP count:  1.121e+16\n",
      "  Optimized FLOP count:  1.920e+08\n",
      "   Theoretical speedup:  58377026.800\n",
      "  Largest intermediate:  3.422e+05 elements\n",
      "--------------------------------------------------------------------------\n",
      "scaling                  current                                remaining\n",
      "--------------------------------------------------------------------------\n",
      "   4                 uvP,ui->ivP                   va,klP,kj,lb,ivP->iajb\n",
      "   4                 kj,klP->jlP                      va,lb,ivP,jlP->iajb\n",
      "   4                 ivP,va->iaP                         lb,jlP,iaP->iajb\n",
      "   4                 jlP,lb->jbP                            iaP,jbP->iajb\n",
      "   5               jbP,iaP->iajb                               iajb->iajb\n"
     ]
    }
   ],
   "source": [
    "print(np.einsum_path(\"ui, va, uvP, klP, kj, lb -> iajb\", Co, Cv, V_df_mp2, V_df_mp2, Co, Cv, optimize=True)[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DF MP2 相关能"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "既然我们已经获得了 DF 近似的 $(ia|jb)$，那么后面的过程与普通的 MP2 并无差异：\n",
    "\n",
    "* $t_{ij}^{ab} = (ia|jb) / D_{ij}^{ab}$\n",
    "\n",
    "* $T_{ij}^{ab} = 2 t_{ij}^{ab} - t_{ij}^{ba}$\n",
    "\n",
    "* $E_\\mathrm{MP2, corr} = T_{ij}^{ab} t_{ij}^{ab} D_{ij}^{ab}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5377072684090779"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_iajb = eri_df_iajb / D_iajb\n",
    "T_iajb = (2 * t_iajb - t_iajb.swapaxes(-1, -3))\n",
    "mp2_df_corr = (T_iajb * t_iajb * D_iajb).sum()\n",
    "mp2_df_corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以看到计算得到的 DF MP2 能量与 PySCF 所给出的 DF MP2 能量几乎完全相等。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.1102230246251565e-16"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mp2_df_corr - mp2_df.e_corr"
   ]
  }
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